I make no apology for returning to the topic of the striking error of physics unearthed by my team of professors, doctors and practitioners of climatology, control theory and statistics. Our discovery the climatology forgot the Sun is shining brings the global-warming scare to an unlamented end. My last article discussing our result attracted more than 800 comments. Here, I propose to answer some of the more frequently-occurring comments, which will be in bold face. Replies are in regular face.

In a temperature feedback loop, the input signal is surface reference temperature before feedback acts. The output signal is equilibrium temperatureE after feedback has acted. The feedback factorf (= 1 – R / E) is the ratio of the feedback responsefE(= E – R) to E. Then E = R + fE = R(1 – f)–1. By definition, E= RA, where A, the system-gain factor or transfer function, is equal to (1 – f)–1 and to E / R.

But your result is too complex. Please state it in simpler terms.

Erroneously, IPCC (2013, p. 1450) defines temperature feedback as responding only to changes in reference temperature. However, feedback also responds to the entire reference temperature. Climatology thus omits the sunshine from its sums and loses the opportunity to find, directly and reliably, the Holy Grail of climate-sensitivity studies – the system-gain factor.

Lacis+ (2010) imagined that in 1850 feedback response accounted for 75% of the equilibrium warming of ~44 K driven by the pre-industrial non-condensing greenhouse gases, implying a feedback factor 0.75, a system-gain factor 4 and an equilibrium sensitivity 4.2 K. i.e., 4 times reference sensitivity 1.04 K (Andrews 2012). Lacis misattributed to the non-condensing greenhouse gases the large feedback response to the emission temperature from the Sun.

In reality, absolute emission temperature in 1850 with no non-condensing greenhouse gases would have been 243.3 K and the warming from those gases 11.5 K, giving a reference temperature of 254.8 K before feedback. The HadCRUT4 equilibrium temperature after feedback was 287.55 K Thus, the system-gain factor, the ratio of equilibrium to reference temperature, was 287.55 / 254.8, or 1.13.

By 2011, if all warming since 1850 was anthropogenic, reference temperature had risen by 0.68 K to 255.48 K. Equilibrium temperature had risen by the sum of the 0.75 K observed warming (HadCRUT4) and 0.27 K to allow for delay in the emergence of manmade warming: thus, 287.55 + 1.02 = 288.57 K.

Climatology would thus calculate the system-gain factor as 1.02 / 0.68, or 1.5. Yet the models’ current mid-range estimate of 3.4 K warming per CO2 doubling implies an impossible 3.25.

In reality, the system-gain factor was 288.57 / 255.48, or 1.13, much as in 1850. It barely changed over the 161 years 1850-2011 because the 254.8 K reference temperature in 1850 was 375 times the manmade reference sensitivity of 0.68 K from 1850-2011. Sun big, man small: nonlinearities in feedback response are not an issue.

Given 1.04 K reference warming from doubled CO2, equilibrium warming from doubled CO2 is 1.04 x 1.13, or 1.17 K, not the 3.4 [2.1, 4.7] K imagined in the CMIP5 models (Andrews, op. cit.). And that, in just 350 words, is the end of the climate scare. There will be too little warming to cause harm.

The feedback-loop diagram simplifies to this black-box block diagram

But your result is too simple. Bringing 122 years of climatology to an end in 350 words? It can’t be as simple as that. Really it can’t. It has to be complicated. Models take account of a dozen individual feedbacks and the interactions between them. IPCC (2013) mentions “feedback” more than 1000 times. Feedback accounts for 85% of the uncertainty in equilibrium sensitivity (Vial et al. 2013). You can’t just jump straight to the answer without even mentioning, let alone quantifying, even one individual feedback. Look, in climatology we just don’t do simple.

Inanimate feedback processes cannot “know” that they must not respond to the very large emission temperature but only to the comparatively small subsequent perturbations. Once it is accepted that feedback responds to the entire input signal, it becomes possible to derive the system-gain factor reliably and immediately. It is simply the ratio of equilibrium to reference temperature at any chosen time. Equilibrium sensitivity to doubled CO2 (after feedback has acted) is simply the product of the system-gain factor and the reference sensitivity to doubled CO2 (before feedback has acted). And that’s that. To find the system-gain factor, one does not need the value of any individual feedback. We can treat the transfer function between reference and equilibrium temperatures simply as a black box.

But each of the five Assessment Reports of the IPCC is thousands of pages long. You can’t just get the answer that has eluded the world’s experts in a few paragraphs.

To quote a former occupier of the office of President of the United States, “Yes We Can.” The “experts” had borrowed feedback math from control theory without understanding it. James Hansen of NASA first explicitly perpetrated the error of forgetting the sunshine in a lamentable paper of 1984. Michael Schlesinger perpetuated it in a confused paper of 1985. Thereafter, everyone in official climatology copied the mistake without checking it. Correcting the error makes it easy to constrain the system-gain factor and hence equilibrium sensitivity.

But climate sensitivity in models is what it is. The science is settled.

All honest experts in control theory will agree that feedback processes in dynamical systems respond to the entire input signal and not just to some arbitrary fraction of that signal. The math is the same for all feedback-moderated dynamical systems – electronic op-amp circuits, process-control systems, climate. Build a test rig. All you need is an input signal, a feedback loop and an output signal. Set the input signal and the feedback factor to any value you like. Now measure the output signal. The circuit doesn’t respond only to some fraction of the input signal. It responds to all of it. We checked by building our own test rig and then getting a government lab to build one for us and to measure the output under a variety of conditions.

Feedback amplifier test circuit built and operated for us by a government lab

But the circuits you built are too simple. Any undergraduate could have built them. You didn’t need to go to a government lab.

We knew official climatology and its devotees would kick and scream and whinge and throw all their toys out of the stroller when they learned of our result. Trillions are at stake. So we checked what did not really need to be checked. Feedback theory has been around for 100 years. To borrow a phrase, it’s settled science. But we checked anyway. Oh, and we went right back to basics and proved the long-established feedback system-gain equation by two distinct methods.

But you didn’t need to prove the equation by two methods. All you needed to do was to prove it by linear algebra.

Yes, indeed. The proof by linear algebra is very simple. Since the feedback factor is the ratio of the feedback response in Kelvin to equilibrium temperature, the feedback response is the product of the feedback factor and equilibrium temperature. Then equilibrium temperature is the sum of reference temperature and the feedback response. With a little elementary algebraic manipulation, it follows that equilibrium temperature is the product of reference temperature and the reciprocal of (1 minus the feedback factor). That reciprocal is, by definition, the system-gain factor.

But we also obtained the system-gain factor as the sum of an infinite series of powers of the feedback factor. Under the convergence condition that the absolute value of the feedback factor is less than 1, the system-gain factor is the sum of the infinite series of powers of the feedback factor, which is the reciprocal of (1 minus the feedback factor), as before. We are guilty of double-checking. Get over it.

Convergence upon the truth

But the equation you use is not derived from any known physical theory.

Yes, it is. See the above answer. But all you really need to know about feedback is that the system-gain factor is the ratio of equilibrium temperature (before feedback) to reference temperature (after feedback). For 1850 and for 2011, we know both temperatures to quite a small margin of error. So we know the system-gain factor, and from that we can derive equilibrium sensitivity to doubled CO2.

But climatology’s version of the system-gain equation is derived from the energy-balance equation via a Taylor-series expansion. It can’t be wrong.

It isn’t wrong. It’s just not useful, because there is much more uncertainty in the delta temperatures than in the well constrained absolute temperatures we use. Neither the energy-balance equation nor the leading-order term in the Taylor-series expansion reliably gives the system-gain factor. It is only when you remember the Sun is shining that you can find the value of that factor directly and reliably.

Climatology in the dark

But if you’re saying climatology isn’t wrong, why are you saying it’s wrong?

Climatology’s system-gain equation, using reference and equilibrium temperature changes rather than absolute temperatures, is a correct equation as far as it goes. It is the difference between two instances of the absolute-value equation. But climatology erroneously limits its definition of feedback as responding only to changes, effectively subtracting out the sunshine. Feedback also responds to the absolute input signal, making it easy to find the system-gain factor and thus equilibrium sensitivity.

No. We have looked out of the window and noticed that the Sun is already Switched On and shining (well, not in Scotland, obviously, but everywhere else). Our calculation starts not with zero Kelvin but with the reference temperature of 254.8 K in 1850. The feedback processes in the climate respond to that temperature and not to any other or lesser temperature. They neither know nor care whether or to what extent they may have existed at any other temperature. They neither know nor care how they might have responded to some other temperature. They respond as they are, and they respond only to the temperature they find. We know the magnitude of the response they engender, for we can measure the equilibrium temperature, calculate the reference temperature and deduct the latter from the former.

But the Earth exhibits bistability. It can have two different temperatures for the same forcing.

Given the variability of the climate, Earth can have several temperatures for a single forcing. But not in the short industrial era. The system-gain factors for 1850 and 2011 are close to identical, indicating that at present there is insufficient inherent instability to disturb our result.

The scrambled account of feedback math in Hansen (1984)

But the feedback system-gain equation is not appropriate for climate sensitivity studies.

Interesting how the true-believers abandon their “settled science” when it suits them. The system-gain equation is mentioned in Hansen (1984), Schlesinger (1985), Bony (2006), IPCC (2007, p. 631 fn.), Bates (2007, 2016), Roe (2009), Monckton of Brenchley (2015ab), etc., etc., etc. If feedback math were not applicable to the climate, there would be no excuse for trying to pretend that equilibrium sensitivity to doubled CO2 is anything like 2.1-4.7 K, still less the values up to 10 K in some extremist papers. As it is, all such values are nonsense anyway, as we have formally proven.

But Wikipedia shows the following feedback-loop block diagram, which proves that feedback only responds to changes, or “disturbances”, in the input signal and not to the whole signal –

A feedback loop diagram from the world’s chief source of fake news

Our professor of control theory trumps the CreepyMedia diagram with the following diagram. And behold, the reference or input signal is at left; the perturbations (in pink) descend from above to their respective summative nodes; and the feedback block (here labeled the “output transducer”) acts on all of these inputs, specifically including the reference signal –

Mainstream block diagram for a control feedback loop

But the models don’t use the system-gain equation. They don’t even use the concept of feedback.

No, they don’t (not these days, at any rate, though until recently their outputs were fed into the system-gain equation to derive equilibrium sensitivity). However, we took some care to calibrate the models’ predicted [2.1, 4.7] K interval of Charney sensitivities using the system-gain equation, which produced exactly the same interval based on the excessive feedback factors derivable from Vial+ 2013. The system-gain equation is, therefore, directly relevant.

The models try valiantly to simulate the multitudinous microphysical processes, many of them at sub-grid scale, that give rise to feedback, as well as the complex interactions between them. But that is a highly uncertain and error-prone method – and even more prone to abuse by artful tweaking than the temperature records themselves: see e.g. Steffen+ (2018) for a deplorable recent example. Besides, no feedback can be quantified or distinguished from other feedbacks or even from the forcings that triggered it by any measurement or observation. The uncertainties are just too many and too large.

Our far simpler and more reliable black-box method proves that the models have, unsurprisingly, failed in their impossible task. By correcting climatology’s error of definition, we have cut the Gordian knot and found the correct equilibrium sensitivity directly and with very little uncertainty.

But you talk of reference and equilibrium temperature when radiative fluxes drive the climate.

Well, they’re called “temperature feedbacks”, denominated in Watts per square meter per Kelvin of the temperature that induced them. They are diagnosed from the models and summed. The feedback sum is multiplied by the Planck sensitivity parameter in Kelvin per Watt per square meter to give the feedback factor. Because the feedback factor is unitless, it makes no difference whether the loop calculation is done in flux densities or temperatures. Besides, our method requires no knowledge of individual feedbacks at all. We find the reference and equilibrium temperatures, whereupon the ratio of equilibrium to reference temperature is the feedback system-gain factor. Anyway, if you want to be pedantic it’s radiative flux densities in Watts per square meter, not fluxes in Watts, that are relevant.

Ten handsome unpersons

But you’re not a scientist.

My co-authors include Professors of climatology, applied control theory and statistics. We also have an expert on the global electricity industry, a doctor of science from MIT, an environmental consultant, an award-winning solar astrophysicist, a nuclear engineer and two control engineers, to say nothing of our pre-submission reviewers, two of whom are the world’s most famous physicists.

But there’s a consensus of expert opinion. All those general-circulation model ensembles and scientific societies and intergovernmental agencies and governments just can’t be wrong.

Yes They Can. In suchlike bodies, totalitarianism prevails (though not for much longer). For them, the Party Line is all, and mightily profitable it is – at taxpayers’ and energy-users’ expense. But the trouble with adherence to the Party Line is that it is a narcotic substitute for independent, rational, scientific thought. The Party Line replace the heady peril of mental exploration and the mounting excitement of the first glimmer of a discovery with a dull, passive, cringing, acquiescent uniformity.

Worse, since the totalitarians who have captured academe ruthlessly enforce the Party Line, they deter terrorized scientists from asking the very questions it is the purpose of scientists to ask. It is no accident that most of my distinguished co-authors now live and move and have their being furth of the dismal scientific establishment of today: for if we were prisoners of that grim, cheerless, regimented, unthinking, inflexible, totalitarian mindset we should not have been free to think the thinkworthy. For these malevolent entities, and the paid or unpaid trolls who mindlessly support them in comments here regardless of the objective truth, punish everyone who dares to think what is to them the utterly unthinkable and then to utter the utterly unutterable. Several of my co-authors have suffered at their hands. Nevertheless, we remain unbowed.

But no one agrees with you.

Here is one of many supportive emails we have had. I get ten supportive emails for every whinger –

“Hi and congratulations on what I believe may have the potential to put the final nail in the coffin of the anthropogenic global warming hysteria. The work of you and your team is very promising and I cannot wait to see how alarmists will go about to attack this. Bring out the popcorn, as we say. The application of feedback theory in this case is simple, physics-wise elegant, mathematically beautiful, and understandable to a wider audience. I am especially excited about how the equation grasps the whole feedback problem without having to deal with all the impossible little details of trying to distinguish which gas does what and without relying on hopelessly complex computer models. And that is why I think it will stick. I will be following this eagerly in the coming months and years and I am considering going to Porto [on 7-8 September: portoconference2018.org: b there or b2] to catch all the latest from others as well, even though your work is the current crown jewel of the anthropogenic global warming debate so far.”

But global temperature is rising as originally predicted.

No, it isn’t –

Our prediction is close to reality: official climatology’s predictions are far out

But you have averaged the two global-temperature datasets that show the least global warming.

Yes, we have. The other three longest-standing datasets – RSS, NOAA and GISS – have all been tampered with to such an extent that they are no longer reliable. They are a waste of taxpayers’ money. We consider the UAH and HadCRUT4 datasets to be less unreliable. IPCC uses the HadCRUT dataset as its normative record. Our result explains why the pause of 18 years 9 months in global warming occurred. Because the underlying anthropogenic warming rate is so small, when natural processes act to reduce warming it is possible for long periods without warming to occur. NOAA’s State of the Climate report in 2008 admitted that if there were no warming for 15 years or more the discrepancy between the models and reality would be significant. It is indeed significant, and now we know why it occurs.

But …

But me no buts. Here’s the end of the global warming scam in a single slide –